$f$-vector inequalities for order and chain polytopes

The order and chain polytopes are two $0/1$-polytopes constructed from a finite poset. In this paper, we study the $f$-vectors of these polytopes. We investigate how the order and chain polytopes behave under disjoint unions and ordinal sums of posets, and how the $f$-vectors of these polytopes are expressed in terms of $f$-vectors of smaller polytopes. Our focus is on comparing the $f$-vectors of the order and chain polytope built from the same poset. In our main theorem we prove that for a family of posets built inductively by taking disjoint unions and ordinal sums of posets, for any poset $\mathcal {P}$ in this family the $f$-vector of the order polytope of $\mathcal {P}$ is component-wise at most the $f$-vector of the chain polytope of $\mathcal {P}$.